Point Group D5h



D5h E 2C5 2(C5)2 5C'2 σh 2S5 2(S5)3 5σv
A'1 1 1 1 1 1 1 1 1
A'2 1 1 1 -1 1 1 1 -1
E'1 2 2cos(2π/5) 2cos(4π/5) 0 2 2cos(2π/5) 2cos(4π/5) 0
E'2 2 2cos(4π/5) 2cos(2π/5) 0 2 2cos(4π/5) 2cos(2π/5) 0
A''1 1 1 1 1 -1 -1 -1 -1
A''2 1 1 1 -1 -1 -1 -1 1
E''1 2 2cos(2π/5) 2cos(4π/5) 0 -2 -2cos(2π/5) -2cos(4π/5) 0
E''2 2 2cos(4π/5) 2cos(2π/5) 0 -2 -2cos(4π/5) -2cos(2π/5) 0
View A
✅
View B


Additional information

Number of symmetry elements h = 20
Number of classes, irreps n = 8
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2C5 2(C5)2 5C'2 σh 2S5 2(S5)3 5σv



Genrate representation from irreducible representations


A'1 A'2 E'1 E'2 A''1 A''2 E''1 E''2



Examples

Iodineheptafluoride Cyclopentadenyl Anion Decaphosphatitanocene Dianion
Heptabromoheptaborane Dianion Pentaprismane Ferrocene (eclipsed)
Fullerene C70


Direct products of irreducible representations


Binary products
⊙ A'1 A'2 E'1 E'2 A''1 A''2 E''1 E''2
A'1 A'1
A'2 A'2A'1
E'1 E'1E'1A'1⊕A'2⊕E'2
E'2 E'2E'2E'1⊕E'2A'1⊕A'2⊕E'1
A''1 A''1A''2E''1E''2A'1
A''2 A''2A''1E''1E''2A'2A'1
E''1 E''1E''1A''1⊕A''2⊕E''2E''1⊕E''2E'1E'1A'1⊕A'2⊕E'2
E''2 E''2E''2E''1⊕E''2A''1⊕A''2⊕E''1E'2E'2E'1⊕E'2A'1⊕A'2⊕E'1

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E'1 A'1⊕E'2E'1⊕E'2A'1⊕E'1⊕E'2A'1⊕A'2⊕E'1⊕E'2A'1⊕2E'1⊕E'2More
E'2 A'1⊕E'1E'1⊕E'2A'1⊕E'1⊕E'2A'1⊕A'2⊕E'1⊕E'2A'1⊕E'1⊕2E'2More
E''1 A'1⊕E'2E''1⊕E''2A'1⊕E'1⊕E'2A''1⊕A''2⊕E''1⊕E''2A'1⊕2E'1⊕E'2More
E''2 A'1⊕E'1E''1⊕E''2A'1⊕E'1⊕E'2A''1⊕A''2⊕E''1⊕E''2A'1⊕E'1⊕2E'2More


Spherical harmonics and Multipoles
Symmetric Powers of Γxyz


Spherical Harmonics Yl / Multipole Symmetric Power [Γl(xyz)]
l 2l+1 Multipole Symmetry Rank [Γl(xyz)]
s (l=0) 1 Monopole A'1 1 A'1
p (l=1) 3 Dipole E'1⊕A''2 3 E'1⊕A''2
d (l=2) 5 Quadrupole A'1⊕E'2⊕E''1 6 2A'1⊕E'2⊕E''1
f (l=3) 7 Octupole E'1⊕E'2⊕A''2⊕E''2 10 2E'1⊕E'2⊕2A''2⊕E''2
g (l=4) 9 Hexadecapole A'1⊕E'1⊕E'2⊕E''1⊕E''2 15 3A'1⊕E'1⊕2E'2⊕2E''1⊕E''2
h (l=5) 11 Dotricontapole A'1⊕A'2⊕E'1⊕E'2⊕A''2⊕E''1⊕E''2 21 A'1⊕A'2⊕3E'1⊕2E'2⊕3A''2⊕E''1⊕2E''2
i (l=6) 13 Tetrahexacontapole A'1⊕2E'1⊕E'2⊕A''1⊕A''2⊕E''1⊕E''2 28 4A'1⊕3E'1⊕3E'2⊕A''1⊕A''2⊕3E''1⊕2E''2
j (l=7) 15 Octacosahectapole A'1⊕A'2⊕E'1⊕2E'2⊕A''2⊕2E''1⊕E''2 36 2A'1⊕2A'2⊕4E'1⊕4E'2⊕4A''2⊕3E''1⊕3E''2
k (l=8) 17 256-pole A'1⊕2E'1⊕2E'2⊕A''1⊕A''2⊕E''1⊕2E''2 45 5A'1⊕5E'1⊕5E'2⊕2A''1⊕2A''2⊕4E''1⊕4E''2
l (l=9) 19 512-pole A'1⊕A'2⊕2E'1⊕2E'2⊕A''2⊕2E''1⊕2E''2 55 3A'1⊕3A'2⊕6E'1⊕6E'2⊕5A''2⊕5E''1⊕5E''2
m (l=10) 21 1024-pole 2A'1⊕A'2⊕2E'1⊕2E'2⊕A''1⊕A''2⊕2E''1⊕2E''2 66 7A'1⊕A'2⊕7E'1⊕7E'2⊕3A''1⊕3A''2⊕6E''1⊕6E''2
n (l=11) 23 2048-pole A'1⊕A'2⊕3E'1⊕2E'2⊕A''1⊕2A''2⊕2E''1⊕2E''2 78 4A'1⊕4A'2⊕9E'1⊕8E'2⊕A''1⊕7A''2⊕7E''1⊕7E''2
o (l=12) 25 4096-pole 2A'1⊕A'2⊕2E'1⊕3E'2⊕A''1⊕A''2⊕3E''1⊕2E''2 91 9A'1⊕2A'2⊕9E'1⊕10E'2⊕4A''1⊕4A''2⊕9E''1⊕8E''2
More

First nonvanshing multipole: Quadrupole

Further Reading

  • A. Gelessus, W. Thiel, W. Weber. J. Chem. Educ. 72 505 (1995)
     Multipoles and symmetry



Ligand Field, dn term splitting


Term symbols for electronic configurations dn
dn Term Symbols
d1 = d9 2D
d2 = d8 1S, 1D, 1G, 3P, 3F
d3 = d7 2P, 2D (2), 2F, 2G, 2H, 4P, 4F
d4 = d6 1S (2), 1D (2), 1F, 1G (2), 1I, 3P (2), 3D, 3F (2), 3G, 3H, 5D
d5 2S, 2P, 2D (3), 2F (2), 2G (2), 2H, 2I, 4P, 4D, 4F, 4G, 6S


Term splitting in point group D5h
L 2L+1 Term Splitting
S (L=0) 1 A'1
P (L=1) 3 A'2⊕E''1
D (L=2) 5 A'1⊕E'2⊕E''1
F (L=3) 7 A'2⊕E'2⊕E''1⊕E''2
G (L=4) 9 A'1⊕E'1⊕E'2⊕E''1⊕E''2
H (L=5) 11 A'2⊕E'1⊕E'2⊕A''1⊕A''2⊕E''1⊕E''2
I (L=6) 13 A'1⊕2E'1⊕E'2⊕A''1⊕A''2⊕E''1⊕E''2

Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement